Question

consider the functions $f(x) = 3x^2$ and $g(x) = -x^3 + 5$, and the functions $h(x)$ and $k(x)$ shown by graphs.

graphs of h(x) and k(x) are included

which of the statements are true? select all that apply.
a. $f$ is neither an even nor odd function.
b. $h$ is an even function.
c. $f$ is an odd function.
d. $k$ is neither an even nor odd function.
e. $g$ is neither an even nor odd function.
f. $g$ is an even function.
g. $h$ is an odd function.
h. $k$ is an odd function.

Explanation:

Step1: Define even/odd functions

A function $f(x)$ is even if $f(-x)=f(x)$ (symmetric about y-axis); odd if $f(-x)=-f(x)$ (symmetric about origin).

Step2: Analyze $f(x)=3x^2$

$f(-x)=3(-x)^2=3x^2=f(x)$. So $f(x)$ is even.

Step3: Analyze $g(x)=-x^3+5$

$g(-x)=-(-x)^3+5=x^3+5$. $g(-x)
eq g(x)$ and $g(-x)
eq -g(x)=x^3-5$. So $g(x)$ is neither.

Step4: Analyze $h(x)$ graph

Graph is symmetric about y-axis. So $h(x)$ is even.

Step5: Analyze $k(x)$ graph

Graph is symmetric about origin. So $k(x)$ is odd.

Answer:

B. $h$ is an even function.
E. $g$ is neither an even nor odd function.
H. $k$ is an odd function.